AF: Small: Fundamental High-Dimensional Algorithms based on Convex Geometry and Spectral Methods

这个项目寻求发现新的算法并开发算法工具来解决算法理论中的基本开放问题，主要有以下几个主题：(1)四舍五入。仿射变换在算法设计和分析中的能力，包括在强多项式时间内求解LP和近似夹住凸体。(2)学习。学习多面体、学习子空间群和识别随机图中种植团的算法。(3)等周不等式。将Cheeger方法推广到高本征值和多重划分，以及KLS超平面猜想。(4)格与凸几何。格上的优化和采样问题，包括：(A)整数规划的复杂性(确定凸体是否与给定的格相交)，(B)割平面方法的复杂性，以及(C)凸体中的格点被有效采样的条件。所讨论的问题是基本的，起源于许多领域，包括优化(离散和连续)、采样、机器学习和数据挖掘。随着高维数据在重要应用领域的可用性越来越高，处理这类数据的高效工具是必要的。该奖项解决了这一需求引发的一些最基本的问题。PI是算法和随机性中心(ARC)的活跃成员，担任其创始主任，并继续与来自不同领域的科学家进行深入合作，以确定可能在理解计算复杂性方面发挥基础作用的问题和想法。该项目将通过在线笔记、教科书和最新调查文章为研究生课程做出贡献。

This project seeks to discover new algorithms and develop algorithmic tools to address fundamental open problems in the theory of algorithms under the following focus topics: (1) Rounding. The power of affine transformations in the design of algorithms and in their analysis, including for solving LP's in strongly polynomial time and approximately sandwiching convex bodies. (2) Learning. Algorithms for learning polyhedra, learning subspace juntas and identifying planted cliques in random graphs. (3) Isoperimetric inequalities. Extensions of Cheeger's method to higher eigenvalues and multi-partitions, and the KLS hyperplane conjecture. (4) Lattices and convex geometry. Optimization and sampling problems over lattices, including: (a) the complexity of integer programming (determining whether a convex body intersects a given lattice), (b) the complexity of cutting plane methods, and (c) conditions under which lattice points in a convex body be sampled efficiently.The problems explored are of a basic nature, and originate from many areas, including optimization (both discrete and continuous), sampling, machine learning and data mining. With the increasing availability of high-dimensional data in important application areas, efficient tools to handle such data are a necessity. This award addresses some of the most basic questions arising from this need. The PI, an active member of the Algorithms and Randomness Center (ARC), served as its founding director and continues in-depth collaborations with scientists from various fields to identify problems and ideas that could play a fundamental role in understanding the complexity of computation. The project will contribute to graduate courses with online notes, textbooks and up-to-date survey articles.